A Magnetically Controllable Valve to Vary the Resistance

of Hydraulic Dampers in Exercise Equipment

A Dissertation Submitted to the Graduate Faculty of

Baylor University

in Partial Fulfillment of the

Requirements for the Degree

of

Master of Science

By

Brett M. Levins

Waco, Texas

August 2005

Copyright © 2005 by Brett M. Levins

All rights reserved

iii

TABLE OF CONTENTS LIST OF FIGURES ......................................................................................................... v LIST OF ABBREVIATIONS.......................................................................................... vi ACKNOWLEDGMENTS ............................................................................................... vii DEDICATION................................................................................................................. viii CHAPTER ONE .............................................................................................................. 1

Introduction.................................................................................................................. 1 CHAPTER TWO ............................................................................................................. 4

Current Adjustable Dampers ....................................................................................... 4 Magnetorheological and Electrorheological Fluids ................................................. 4 Restrictive Orifice Designs ...................................................................................... 5

CHAPTER THREE ......................................................................................................... 7 Presented Adjustable Damper Model .......................................................................... 7

Description of Simple Damper ................................................................................ 7 Description of Presented Damper ............................................................................ 9 Analysis of the Magnetic Valve............................................................................... 10

CHAPTER FOUR............................................................................................................ 17 Experiment Results and Conclusions .......................................................................... 17

Effective Orifice Area Experiments......................................................................... 17 Effective Orifice Area Experiment of Clear Cylinder ......................................... 17 Clear Cylinder Effective Orifice Area Experiment Results................................. 18 Effective Orifice Area Experiment of Testing Piston.......................................... 22 Testing Piston Effective Orifice Area Experiment Results ................................. 25 Testing Piston Effective Orifice Area Experiment Results ................................. 25

Magnetic Strength Experiments............................................................................... 29 Description of Magnetic Strength Experiment .................................................... 29 Clear Cylinder Magnetic Strength Experiment Results....................................... 30 Steel Cylinder Magnetic Strength Experiment Results ....................................... 30

Comparison of Steel Cylinder and Clear Cylinder Data.......................................... 33 Conclusions.............................................................................................................. 34

APPENDICES ................................................................................................................. 36 APPENDIX A.................................................................................................................. 37

MATLAB Code – Gap data......................................................................................... 37 Readin.m .................................................................................................................. 37 process.m ................................................................................................................. 38 convert.m ................................................................................................................. 39 get_ext_stroke.m...................................................................................................... 40 ext_stroke.m............................................................................................................. 41 avals.m ..................................................................................................................... 42 geta.m....................................................................................................................... 43 Qvals.m .................................................................................................................... 44 plotFV.m .................................................................................................................. 46 plotall.m ................................................................................................................... 48

iv

stdev.m ..................................................................................................................... 49 APPENDIX B .................................................................................................................. 50

MATLAB Code – Force-velocity Data from Applied Current ................................... 50 readin.m ................................................................................................................... 50 process.m ................................................................................................................. 51 convert.m ................................................................................................................. 52 get_ext_stroke.m...................................................................................................... 53 ext_stroke.m............................................................................................................. 54 avals.m ..................................................................................................................... 55 geta.m....................................................................................................................... 57 FI_plots.m ................................................................................................................ 58 FI_plots_mm.m........................................................................................................ 59 plotall.m ................................................................................................................... 60 separate_extension_plots.m ..................................................................................... 61

APPENDIX C .................................................................................................................. 64 LabVIEW Data Acquisition Code ............................................................................... 64

MagThesis.vi Front Panel ........................................................................................ 64 MagThesis.vi Block Diagram .................................................................................. 65 hkWriteLong.vi Front Panel .................................................................................... 71 hkWriteLong.vi Block Diagraml ............................................................................. 71

APPENDIX D.................................................................................................................. 72 HyperKernel C Code ................................................................................................... 72

CurvesDevelop.c ...................................................................................................... 72 APPENDIX E .............................................................................................................. 77 Force-Velocity Data for Gap Experiment.................................................................... 77

APPENDIX F................................................................................................................... 81 Force-Velocity Data for Applied Current Experiment ................................................ 81

v

LIST OF FIGURES

Figure 1: Sample Orifice Sizes for Simple Piston Head................................................. 8 Figure 2: Presented Damper Diagram............................................................................. 11 Figure 3: Conceptual Valve Assembly ........................................................................... 12 Figure 4: Actual Piston Assembly .................................................................................. 12 Figure 5: Testing Set-Up for Clear Cylinder .................................................................. 19 Figure 6: Sample Force-Velocity Data for 0.25mm Gap................................................ 20 Figure 7: Q Versus Gap Size for Clear Cylinder. ........................................................... 20 Figure 8: Clear Cylinder Effective Orifice Area............................................................. 22 Figure 9: Piston Head for Gap Testing ........................................................................... 23 Figure 10: Testing Lever Assembly................................................................................ 23 Figure 11: Close-up of Piston when Mounted on Lever Assembly................................ 24 Figure 12: Force-Velocity Data for 0.27mm Gap........................................................... 28 Figure 13: Force-Velocity Data for 1.20mm Gap........................................................... 28 Figure 14: Q Versus Gap Size for Steel Cylinder........................................................... 29 Figure 15: Force-Velocity Characteristics from Applied Current for Clear Cylinder.... 32 Figure 16: Sample Force-Velocity data for the Magnetic Strength Experiment ............ 32 Figure 17: Force-Velocity Characteristics from Applied Current for Steel Cylinder .... 33

vi

LIST OF ABBREVIATIONS

x magnetic valve gap; the gap between the magnet and the piston head Ain fluid inlet orifice area (constant) Aout fluid outlet orifice area (constant) Ap area of piston head (constant) Amax max effective fluid orifice area (constant) A(x) measured effective fluid orifice area

)(xA∧

projected effective fluid orifice area fflu force of fluid exerted on valve fmag magnetic force exerted on valve fgas gas accumulator force exerted on piston ∆P pressure difference across piston F force developed by damper ρ fluid mass density (constant) k adjustment to Bernoulli’s equation (constant) vp velocity of piston Q(x) stiffness of damper Qmin minimum stiffness of damper g(x) fluid force modulation function od orifice diameter (constant) I applied current

vii

ACKNOWLEDGMENTS

I would like to thank Dr. Ian Gravagne for his exceptional teaching and guidance

throughout my research at Baylor University. Also, I am grateful for a grant from Curves

International (contract 097-04-Curves) to fuel my research. Finally, I would like to show

appreciation to Ashley Orr and Blake Branson for technical assistance in construction and

assembly of the dampers.

viii

DEDICATION

To my parents, whose constant support continues to be my motivation.

1

CHAPTER ONE

Introduction

It has become quite noticeable in recent years that linear fluid damping can be

utilized as an effective and economical way to provide resistance for exercise machines.

Linear dampers have numerous advantages to recent counterparts such as weights and

resistance bands. Dampers are small and compact, unlike commonly employed resistance

devices that need cables and pulleys to provide resistance. Also, fluid dampers are mass

produced for the automotive industry, leaving them inexpensive and easy to obtain. The

most observable advantage over other types of resistances, however, is the ability to

provide resistance in both compression and extension (both directions of the stroke), often

referred to as “double positive” resistance. Exercise machine designers can utilize linear

dampers to provide exercise to several different muscle groups on the same machine. This

is highly beneficial to exercise companies that wish to provide circuit training with a small

number of machines. In addition to smaller numbers of machines, dampers aesthetically

provide exercise equipment with an appearance that is not overwhelming; a positive for

many beginner exercisers who may be discouraged from a complex weight and pulley

system.

Despite the fact that linear dampers have many advantages, they do have one

disadvantage: they are difficult to adjust. Adjustability, though sometimes found on

pneumatic dampers, is rarely found on fluid dampers due to the fact that fluid dampers

must remain sealed at all times and do not have the ability to draw in or release air from

the damper. There has been research into linear dampers not based on fluid or gas [5][9],

2

but without the damping effect of fluid, energy must be dissipated through the active

element of the design. This active element is commonly an electromagnet or a motor, and

tends to increase the size and complexity of the device.

In the past, the fluid adjustability problem has been remedied in two different

ways. The first solution is to physically change the size of the orifice using a knob or

similar device. This technique is commonly used in gas dampers, but not in fluid

dampers. Since fluid dampers need to be completely sealed to prevent air from entering

the system, it is difficult to design dampers with mechanical fixtures that do not jeopardize

the damper seal. It is also possible to adjust orifice sizes within the cylinder by

controlling a valve using different types of motors. This valve controlling tends to add

complexity to the system and requires feedback from the motor, leaving the damper

difficult to control.

Conversely, the second way to control a fluid damper is to actually change the

viscous properties of the fluid itself. Recently, the use of magnetorheological (MR) and

electrorheological (ER) fluids have increased in damper design [1] [2] [4] [6] [7]. These

fluids have the ability to become more viscous upon the application of a magnetic or

electric field, thus dynamically adjusting the resistance of the damper depending on the

strength of the field. These types of dampers appear to be the best choice, but the high

cost of the fluid leaves the dampers as a non-economical solution to many exercise

companies. Each of these control designs will be examined in more depth later.

The author was approached by an exercise company providing guidelines to the

development of an adjustable damper that gave purpose to this thesis. An adjustable

damper needed to be developed for the use in exercise equipment that would effectively

3

change resistances to accommodate for a wide force range. The damper needed to be a

simple design that would be easy to maintain, and also have the ability to be mass

produced at a minimal cost. The author decided that the most suitable way to control the

strength of the damper at a minimal cost would be to design for an internal variable

orifice. This thesis presents a damper design that will control orifice size in an

unconventional manner. The damper utilizes two electromagnets and their repulsive

magnetic field to block and control the size of an orifice by creating a magnetic valve.

The author presents a hypothesis of the behavior of the magnetic valve suggesting how the

effective orifice area on the piston head is related to the gap distance between the magnet

and the piston head. Using a developed testing platform, force-velocity tests have been

performed and presented to describe the characteristics of two constructed cylinders.

Finally, the force-velocity characteristics of the developed dampers are exhibited and

analyzed for a range of applied currents to the electromagnets.

4

CHAPTER TWO

Current Adjustable Dampers

Magnetorheological and Electrorheological Fluids

As mentioned earlier, there are currently two different ways to dynamically control

the resistance of a fluid damper. The most popular way is to incorporate MR or ER fluid

into the cylinder. MR fluid consists of a base fluid, usually hydraulic oil, and suspended

particles of iron randomly dispersed within the fluid. When a magnetic field is applied

across the fluid, the iron particles line up and create “chains” along the magnetic field

lines. These chains, in turn, resist the flow of the fluid. The strength of the chains, and

resulting flow resistance, depends on the strength of the magnetic field [4]. Similarly, ER

fluids also consist of a base fluid and suspended particles. They differ from MR fluids in

that the particles form chains when an electric field is applied directly to the fluid itself.

Both fluids have proven to be exceptionally strong and have large bandwidths. MR fluids

are known to resist pressures of 50-100 kPa at maximum magnetic field strengths of 250

kA/m, while ER fluids can resist pressures of 2-5 kPa at electric field strengths of up to 4

kV/mm [10]. The large voltages that ER fluids require tend to guide engineers to the use

of the more manageable MR fluids. The main problem with MR and ER fluids is the fact

that they begin to break down over time. This phenomenon is known as “in-use

thickening [11].” When an ER or MR fluid is subjected to high shear stresses for a

significant period of time, the fluid begins to thicken until it eventually becomes

unmanageable. In typical exercise machines, it is projected that a damper will incur a

million strokes per year and needs the ability to last around three to five years. Currently

5

produced MR and ER fluids have the ability to withstand these numbers of strokes, but

only when high priced agents are added to the fluid. This high price (around $100 per US

liter) leaves MR or ER fluid as a bad economical choice for exercise equipment.

Additionally, the agents added to commercially produced fluid not only improve the

lifetime of the fluid, but also reduce the unavoidable problem of particle settlement. For

this reason, simpler MR or ER fluids are unable to be developed for this application due to

the fact that the particles will begin to settle between uses of the equipment.

In order to avoid the use of excessive volumes of MR fluids in dampers (and

associated extra cost), MR sponge devices have been developed and implemented in

certain applications [2]. MR sponges consist of a piston head with an electromagnet

covered with a sponge like material that has been soaked in MR fluid. Once current is

applied to the electromagnet, the MR sponge on the piston head acts in direct-shear mode

with the piston wall. This is in contrast to the pressure driven flow mode MR fluid

experiences when flowing through an orifice [4]. Since the MR fluid is soaked into the

sponge device on the piston head, no fluid is needed inside the damper. Therefore, the

damper can function with air in the volume usually occupied by the base oil of damper.

The smaller amount of MR fluid significantly lowers the cost of the damper. MR fluid

sponge devices were developed and tested for application in this project. Although the

cost of the damper was manageable for large scale production, necessary force levels for

use in exercise equipment could not be reached by the developed MR sponge dampers.

Restrictive Orifice Designs

A second way to change a damper’s resistance is to restrict the size of the orifice

either externally or internally. External control of an adjustable orifice typically consists

6

of a knob on the outside of the damper that dynamically controls the size of the orifice.

The knob can be turned to open or close the orifice with the precision of the change in

orifice size depending on the thread size of the knob. This type of design is usually only

implemented on air dampers due to the importance of sealing fluid dampers [3]. Also,

external control is often awkward and time-consuming to change, another disadvantage

for exercise machines. Fluid dampers with mechanical adjustments are rarely used in

exercise equipment; refer to [8] for an implementation for this type of damper.

Internal control of orifice size is usually accomplished by a motor-like device on

the piston head that has the ability to block an orifice. These types of constructions are

beneficial because they are completely contained within the cylinder and have shown to

work relatively well. The main problem with motor type designs is that they require

feedback for the controller. Feedback often complicates systems and requires extra

hardware for control. The more control that is needed for the system, the more expensive

the system becomes.

7

CHAPTER THREE

Presented Adjustable Damper Model

An adjustable damper focusing on a simple design has been developed and is

presented in the following chapter. Work described and presented in this paper closely

follows work presented by Levins and Gravagne [12].

Description of Simple Damper

Before analyzing how the presented damper works, we must first understand, in

general, how a typical damper works. A simple damper consists of two volumes of fluid

separated by a piston head that contains an orifice (a hole smaller than the diameter of the

piston head that allows fluid to flow between the two volumes). Upon moving either

direction, a pressure differential is created between the two volumes and causes the fluid

to flow through the orifice in order to equal the pressure once again. Since the area of the

orifice is smaller, usually significantly smaller, than the area of the piston head, the

movement of the piston head is hampered. This hampering of the piston head is often

referred to as inertial damping. Imagine a small mass of fluid on the high pressure fluid

volume. The small mass is forced from the high pressure volume into the orifice. Since

the orifice contains significantly less volume than the original fluid volume, the mass will

travel at a faster velocity while inside the orifice. Once the mass travels through the

orifice and reaches the larger fluid volume on the other side of the piston, the mass will

once again slow down. This process of speeding up and then slowing down causes the

damping effect “felt” in the damper.

8

The amount of force required to displace a piston head at a given velocity for a

simple orifice is shown in figure 1. Note that the force-velocity (FV) curve for an orifice

is quadratic with the stiffness increasing as the size of the orifice becomes smaller. The

fact that the faster a damper moves, the harder it becomes is beneficial to exercise

equipment. This helps to account for a wide range of user strengths on an exercise

machine. If a stronger exerciser is on the machine, they can simply move the damper

faster and obtain the force levels they need out of the damper.

Figure 1: Sample Orifice Sizes for Simple Piston Head – The force-velocity characteristics of a simple damper and piston head for 3 different orifice sizes. The plot demonstrates the fact that as orifice sizes begin to decrease, the damper becomes stiffer.

9

However, exercising at faster speeds begins to create more of an aerobic workout

than a strength workout, a non-ideal result for some exercise companies. Therefore, a

damper that can control the size of the orifice on the piston head would be beneficial in

order to account for a range of strength for different exercisers.

Description of Presented Damper

The presented damper model is shown below in figure 2. The damper consists of

three main chambers: a magnet chamber and two fluid chambers. The two fluid chambers

are represented by the volume of fluid in the chamber, VA and VB. The magnet chamber is

located between two piston heads, Piston A and Piston B, which are roughly 10 cm apart

and divide the fluid volumes from the magnet chamber. A gas accumulator is included

within the piston in order to account for the difference in volume created by the piston rod

during a stroke.

The piston heads consist of one orifice and a check valve that allows fluid to only

flow from within the magnet chamber to the fluid chamber. The orifices on each piston

head are equipped with a rubber washer on the magnet chamber side to guarantee that

there is a strong seal between the associated magnet and piston head. Each piston head

contains a Teflon seal between the piston head and the inside wall of the damper to ensure

all fluid must travel through the orifice or check valve instead of escaping around the sides

of the piston head. Inside the magnet chamber are two electromagnets, Magnet A and

Magnet B, that are capable of sliding separately and without restraint on a steel shaft.

Once excited, the magnets are oppositely poled and consequently repel each other,

creating a small gap between the two.

10

During an extension stroke, a static pressure difference is created between VA and

VB. This pressure difference causes fluid to flow from VA through the orifice on Piston A

into the magnet chamber. In order for this to happen, Magnet A must displace off of

Piston A to allow the fluid to enter the magnet chamber. The entering fluid then exits out

of the check valve on Piston B. Note that Magnet B is blocking and sealing the orifice on

Piston B, therefore all the exiting fluid must travel through the check valve. This process

is reversed for a compression stroke. Fluid travels into the orifice on Piston B, through

the magnet chamber, and out of the check valve on Piston A. Figures 3 and 4 show an up

close version of the conceptual design of the piston head assembly and the actual

construction of the piston head assembly, respectively.

Analysis of the Magnetic Valve

Understanding the way in which a magnet interacts with the orifice on the

piston head is crucial to the analysis of this damper. Jets of fluid flowing through the

orifice on the piston head strike the face of the magnet upon entering the magnet chamber.

The magnet on the high pressure side of the piston head acts as a valve in which the gap

between the magnet and the piston head controls how much fluid is entering the magnet

chamber. The valve controls how much fluid is entering the magnet chamber by

restraining the area of the inlet orifice, Ain. Therefore, Ain is controlled by the strength of

the magnetic field (the controlling factor on the gap size). However, there is a maximum

rate that fluid can enter the magnet chamber which depends solely on the maximum area

of the orifice, Amax; a value determined by the orifice diameter. Imagine spraying water

from a hose against a brick wall. As the spray becomes closer to the wall, the exiting

water feels more forceful against the wall. A common way to re-create that same close

11

proximity force would be to back up from the wall and obstruct the outlet of the hose.

This obstruction has created a smaller orifice. This implies a correlation involving the gap

size between an orifice and an impeding structure, and the effective size of that orifice.

Figure 2: Presented Damper Diagram – A diagram of the presented damper body and valve. The remote reservoir of the system is represented by the accumulator in this figure for demonstration purposes.

12

Figure 3: Conceptual Valve Assembly – The conceptual design of the magnetic valve when the 2 electromagnets are at rest. If excited with current, the gap would be visible between the magnets.

Figure 4: Actual Piston Assembly – A close-up of the magnetic valve on the piston heads after it has been constructed.

13

In the presented damper design, the piston head consists of a single orifice and a

check valve. The check valve is significantly larger than the orifice to allow the FV

characteristics of the damper to be dependant on the orifice size alone. Therefore, the

orifice shows a maximum inlet area Amax of,

2

max 2⎟⎠⎞

⎜⎝⎛= do

A π (1)

where od is the orifice diameter.

Now, let x(t) represent the gap between the piston head and the magnet

( )mmx 4.10 ≤≤ , m be the mass of the magnet, and b the fluid damping constant. While

in motion, the fluid force exerted onto the magnet, fluf , must equal the magnetic field

force, magf , as

0=+−+ magflu ffxbxm &&& (2)

The magnetic force, magf , is controlled by the applied current I. Therefore, since the

magnets essentially adjust the size of the orifice, the presented damper becomes

dynamically adjustable depending on the amount of current that is applied to the magnets.

Increasing currents will create a stronger repulsive magnetic field between the magnets

which, in turn, will allow the magnets to resist a larger force from the fluid trying to enter

into the magnet chamber. An interpretation of this would be that the magnet is

“shrinking” the size of the orifice. Also, since the values of x stay below 1mm, the design

takes advantage of the fact that the strength of opposing magnetic fields increase

exponentially as two magnets become closer in proximity. The small x values are also

beneficial because this leaves ( )Ifmag independent to the value of x. On the other

14

hand, fluf is a function of x and the differential pressure across the piston head,

AB PPP −=∆ .

The formulation of ( )Pxf flu ∆, begins to become complex because it must capture

the transition from a pure reaction force (when x = 0 and fluf is only dependant on the

pressure) to an impulse force (when the jets of fluid from the orifice impinge on the face

of the magnet as the gap opens). This implies that fluf depends on the velocity of the fluid

jets; furthermore, the fluid jet velocity depends on the effective orifice size and the

differential pressure across the piston head. Therefore, it is observed that

)(max xgPAf flu ∆= (3)

with the arbitrary function 1)( ≤xg . It is beyond the scope of the thesis to fully

investigate the function g(x). Now we turn our attention to the effective orifice area as a

function of the gap, x.

Since the acting magnetic valve only changes the effective orifice size, Bernoulli’s

equation supplies the suitable relationship. Now, let )(xA be the effective orifice area of

the piston head, noting that max)( AxA → as ∞→x . Also, if we assign the piston head

area to be pA and assume that the cross-sectional area of the piston rod is small compared

to the area of the piston face, Bernoulli’s equation gives

222

2

)(2 pp vxAk

AP

ρ=∆ (4)

In true laminar flows, the constant k would equal 1, although in practice it lies in the range

of 0.8 to 0.9 [3]. The force of the damper is related to the P∆ of the system by the

aggregate damper force, pPAF ∆= . Also, when relating the velocity to the force, two

15

additional cylinder effects arise: the velocity-dependent friction of piston head and rod

when sliding within their respective seals, and the force exerted by the gas accumulator.

Bernoulli’s equation now becomes

gasppp

p fcvvxAk

AF ++= 2

22

3

)(2ρ

(5)

For the presented damper, the diameter of the piston rod is only 12.7cm. This diameter

value makes the area difference between the magnet chamber side and fluid volume side

of a piston head is less than 8%. Therefore, we can assume the accumulator

force 0≅gasf . Also, scatter plots show that friction is negligible for most gap sizes and a

quadratic representation is more than adequate to model the data. This implies that 0≅c .

Friction details will be discussed further in the results portion of the thesis.

We now turn our attention to the main focus of this chapter: the effective orifice

area function, A(x). A hypothesis for A(x) is developed by modifying equation (1) to

incorporate a function of the gap size, x. We assume that the fluid “sees” the orifice as a

virtual cylinder. The cylinder has a radius of 2do

and a height of x. Therefore, the

cylinder area )*2

(2 xo

A din π= . As the gap widens, the cylinder surface area increases

linearly until the limiting maxA function is reached and increases in gap no longer allow for

greater fluid flow. Now, we have an hypothesis function

( ){ }max,min)( AxAxA in=∧

(6)

It is important to notice that max)( AxA →∧

as ∞→x . This implies that once the gap

becomes significantly larger than the orifice diameter, the magnet is no longer acting as a

16

valve and the orifice is the sole limiter to the flow of the fluid. The next section will

present an experiment and data results to justify the above hypothesis.

17

CHAPTER FOUR

Experiment Results and Conclusions

In this chapter, we want to experimentally examine the effective orifice area, A(x),

and the force-velocity characteristics of the damper as a function of the applied current,

F(I). Separate experiments were conducted for each query and presented within the

chapter. The presented results are for two separate damper designs. Initially, a damper

was constructed with a cylinder wall of a clear plastic for viewing purposes. This

damper, referred to as the “clear” damper, was previously presented by Levins and

Gravagne [12]. Upon reviewing the results of the clear damper, a new damper was

constructed with a high tolerance steel cylinder wall in order to relieve noisy data in the

system. This cylinder is referred to as the “steel” cylinder. The gap experiment data for

each cylinder is presented initially, followed by the applied current experimentation

results.

Effective Orifice Area Experiments

Effective Orifice Area Experiment of Clear Cylinder

Testing for the clear cylinder was conducted on an exercise machine. The stroke

data was recorded using a 500 lb. tension/compression load cell (Transducer

Technologies SSM-500 with calibrated signal conditioner) and an externally mounted

digital position sensor (Unimeasure LX-EP-15) for the force and linear

displacement/velocity measurements, respectively. Figure 5 shows the testing set-up of

the clear cylinder.

18

For this experiment, the piston head contained 3 orifices of 2.68mm in diameter.

Therefore, Amax consists of 3 inlet areas in “parallel” with each other, and in series with

the outlet check valve orifice,

2

268.23 ⎟

⎠⎞

⎜⎝⎛= πinA ;

2

222.3

⎟⎠⎞

⎜⎝⎛= πoutA

49.5max =+

=outin

outin

AAAA

A mm2 (7)

Thus, the hypothesis of A(x) modifies equation (7) by assuming the total cylinder surface

area is ( ) xx ππ 834.123 ≅×× . Therefore, the hypothesis function

( ) { }{ } outin

outin

AAxAAx

xA+×

=,8min,8minˆ

ππ

(8)

Clear Cylinder Effective Orifice Area Experiment Results

The gap, x, was fixed at 5 different positions,

{ }14.1,02.1,76.0,51.0,25.0∈x millimeters (9)

For each gap size, force-velocity data was gathered and plotted with a least square error

best fit curve. Figure 6 shows a sample of the scatter plot data and best fit curve for the

gap experiment. The quadratic stiffness coefficients for the cylinder gap experiment

were found to be

( ) { } 322

2

10116,117,116,136,195)(2

×∈=xAk

AxQ pρ

(10)

19

Figure 5: Testing Set-Up for Clear Cylinder – View of the clear cylinder mounted on the testing machine.

20

Figure 6: Sample Force-Velocity Data for 0.25mm Gap – Scatter plot data for the compression stroke of the clear cylinder with a 0.25mm gap.

Figure 7: Q Versus Gap Size for Clear Cylinder – plot of Q(x) showing measured values and least square best fit curves. Vertical error bars indicate one standard deviation.

21

This data suggests that Q(x) approaches a minimum value, Qmin. This is easily

predicted if we recall that max)( AxA → as ∞→x . The piston head for this design had a

diameter of 47.5mm, and the hydraulic fluid used in the experiment had a mass density of

approximately 900 kg/m3. Using k = 0.9, it was found that Qmin = 107.2 * 103. Q(x) is

plotted in figure 7 along with several different modeling functions. The modeling

function of 2

1x

showed to fit the data accurately and resulted in a ( ) min2

0058.0 Qx

xQ +=

best fit curve.

Rearranging equation (10) for the valved orifice area function, A(x), gives

0058.02

)( 2min

2

2

3

+=

xQx

kA

xA pρ (11)

The effective valved orifice area is plotted in figure 8 against the hypothesis function.

The plot showed that the hypothesis function is fairly accurate in its prediction. Due to

accuracy of the hypothetical applied orifice function to the experimental data of the clear

cylinder, the same hypothesis was utilized to analyze the steel cylinder.

Finally, figure 7 illustrates the fact that only a small gap is needed between the

magnets to achieve an efficient valve. The effective orifice area increases over 90%

within the first 0.5mm of movement by the magnetic valve. Note that this is highly

important due to the fact that magnetic field strength increases the closer the magnets are

in proximity.

22

Figure 8: Clear Cylinder Effective Orifice Area – comparison of the measured effective orifice area function and the hypothesis function. Effective Orifice Area Experiment of Testing Piston

In order to further demonstrate the variable orifice phenomenon, a simpler,

separate piston was designed and tested to allow for more variability in gap size. The

piston, shown below in figure 9, consists of a piston head with a single orifice and a fluid

blocking surface. The blocking surface is threaded and attached to the piston head by a

bolt. This design allowed for testing of a wider range of gap distances. The piston was

tested using the lever assembly shown in figure 10 and used the same previously

described force and displacement/velocity sensors. Figure 11 shows a close-up of the

damper body when it is in the lever assembly.

23

Figure 9: Piston Head for Gap Testing – The piston head assembly for the gap testing. The assembly consists of a piston head with a single orifice and a fluid blocking surface. The fluid blocking surface is threaded and connected with a bolt, allowing for the gap size to be easily adjusted.

Figure 10: Testing Lever Assembly – The lever assembly used for gathering test data. The lever assembly consists of a fixed base, post, and lever. Two plastic bearings connect the post and lever allowing for low friction, vertical movement of the lever. The piston connects to the base and lever when testing.

24

Figure 11: Close-up of Piston when Mounted on Lever Assembly – The damper when mounted on the lever assembly with the force and displacement sensors used to collect the testing data.

25

Testing Piston Effective Orifice Area Experiment Results The experimental piston was tested at 6 different gap distances,

{ }2.1,84.0,60.0,45.0,36.0,27.0∈x millimeters (12)

The experimental piston head has only one orifice with a diameter of 2.49mm instead of

three orifices like the piston head of the clear cylinder. This leaves Amax to be calculated

as,

22

max 87.4249.2 mmA =⎟

⎠⎞

⎜⎝⎛= π (13)

This diameter size also presents the hypothesis cylinder area from equation (6) to be

xxAin ππ 5.2)*25.1(2 ≅= mm2. For each of the gap distances, force-velocity scatter

plots were obtained through the experimental setup previously described, and least square

error best-fit curves were determined to give quadratic stiffness coefficients for equation

5 as

( ) { } 322

2

102.97,5.97,100,105,123,145)(2

×∈=xAk

AxQ pρ

(14)

Force-velocity scatter plots for the gap sizes of 0.27mm and 1.2mm are shown in

figures 12 and 13, respectively. The scatter data is plotted with the best fit curve

corresponding to the gap size. It is important to be aware that a linear element was added

to the best fit curve in the larger gap size in order to fit the data better. This linear

element is due to the friction between the piston rod and the top seal of the damper as the

rod is moving. This friction is not seen in the smaller gap size plot because the force

required to displace the fluid through the smaller orifice is so much larger than the

frictional force that the fluid force overshadows the frictional force. Without the

26

frictional force, the cylinder behaves in the quadratic nature one would expect. The

damper was only tested for larger gap sizes in order to see if the effective orifice area

would, in fact, approach Amax. Therefore, since the friction was not prevalent in the gap

data that was most crucial to the modeling of the damper, the velocity dependant friction

in equation 5 was able to be disregarded.

Also notice in figure 12, the scatter plot of the smaller gap size, that a constant

force offset needed to be added to fit the data. Since the smaller gap size represents a

smaller orifice, the magnetic valve is able to resist a larger static pressure before

displacing from the piston head and allowing the piston to move.

Some ambient data points can be seen in figure 12 as well. These data points can

be attributed to pressures in the remote reservoir. The points represent a stroke of the

damper but do not correspond to the actual characteristics of the damper. The remote

reservoir of the shock was filled with 120 psi of air. When the differential pressure

across the piston heads during a stroke exceeded the reservoir pressure, the air within the

reservoir was compressed. The compression of the air in the remote reservoir of the

damper caused the piston to spring back and create the “trampoline points” seen in the

plot. The pressure in the reservoir prompted the piston to move at a high velocity with a

low force, as shown by the trampoline points in the plot. The reservoir continued to push

the piston until enough fluid passed through the orifice to allow the pressure in the

reservoir to once again exceed the differential pressure on the piston heads. At this point,

the data points once again tracked the true characteristics of the damper. In order to

minimize the amount of trampoline points in the scatter plot data, the damper was held

27

after a compression stroke until the pressure in the reservoir was once again larger than

the differential pressure of the fluid volumes.

Plots of each of the Q values against the gap distance, as well as the least square

best-fit line for Q(x), are shown in figure 14. The plot also indicates that Q(x)

asymptotically approaches a lower bound, Qmin, much like the Q(x) for the clear cylinder.

The experimental piston head is 45.72mm in diameter, and, using a graduated cylinder

and digital scale, the density of the fluid was calculated to be 843 kg/m3. By allowing k

= 0.9, it was determined that Qmin = 97.23 x 103. This value of Qmin agrees well with the

asymptotic approach of the modeling equation for Q(x).

Figure 14 also contains the hypothesis function of Q(x). Notice how the

hypothesis function increases much more rapidly than the measured function. This can

be attributed to a number of factors. First, if the piston head does not fully seal with the

inside of the damper body, fluid will escape around the outside of the piston head instead

of flowing solely through the orifice. This effect will cause the Q values to flatten out

much like on the figure. This makes sense, intuitively, because the extra fluid escaping

would represent a larger orifice size. In turn, a larger orifice size would represent a

smaller Q value.

Secondly, due to the rod sliding past the top seal, a small amount of velocity

dependent friction is implemented into the system. This small amount of friction would

introduce a linear component into the modeling equation that would also aid in flattening

out the data. Any combination of these two factors could contribute to the slight

discrepancy between the hypothetical behavior and the actual behavior of the damper.

28

Figure 12: Force-Velocity Data for 0.27mm Gap – Scatter plot data and corresponding least square best fit curve for 0.27mm gap testing.

Figure 13: Force-Velocity Data for 1.20mm Gap – Scatter plot data and corresponding least square best fit curve for 1.20mm gap testing.

Trampoline Points

29

The important aspect to take from figure 14 is the fact that the experimental data

does follow the relationship we expect. The Q values exponentially decrease with larger

gap sizes and begin to approach the Qmin value that represents the maximum orifice area,

Amax.

Figure 14: Q Versus Gap Size for Steel Cylinder – Comparison of the calculated Q values and corresponding least square best fit curve to the hypothetical Q(x) function presented in the chapter. Vertical error bars indicate one standard deviation.

Magnetic Strength Experiments

Description of Magnetic Strength Experiment

In the previous experiment, the gap size in between the magnets was constrained

in order to explore the activity of the magnetic valve. In this experiment, we allow the

magnets to freely move and vary the current applied to each of the magnets. As

30

mentioned before, the amount of applied current controls the strength of the magnetic

field, which, in turn, controls the regulation of the gap size for the magnetic valve. Also

as in the gap experiment, an exercise machine was used to collect force-velocity data for

the clear cylinder and the lever assembly was used to collect the force-velocity data for

the steel cylinder.

Clear Cylinder Magnetic Strength Experiment Results Force-velocity data was recorded for 8 different current values,

{ }2600,2260,1930,1540,1150,770,380,0∈I milliamps (15)

Figure 15 shows the second-order best fit polynomials of the compression and extension

strokes for the 8 different current values. The scatter plot data was omitted for

readability. For this damper, the resistance increases 107% at a speed of 50 mm/s when

2.6A of current is applied. Notice that there is no constant offset for the polynomials on

this plot. This behavior is unexpected and insinuates that the damper is acting more like

the model of the theoretical damper instead of exhibiting common difficultly-predicted

characteristics of dampers.

Steel Cylinder Magnetic Strength Experiment Results Force-velocity data for the damper was gathered for a series of applied currents,

{ }2083,1823,1563,1302,1042,781,521,260,0∈I milliamps (16)

For each set of force-velocity data, a second order least-square best fit function was

conformed to the raw data. Figure 16 is an example of the raw force-velocity data with

its corresponding best fit curve. The previously described trampoline points are also

present in the magnetic strength experiment results of the steel cylinder.

31

Although the trampoline points are present in the steel cylinder data, the true

characteristics of the cylinder are easily visible. The discernible cylinder characteristics

allow the trampoline points to be ignored. Also, for plots of the lower current values,

{ }781,521,260,0∈I , a linear variable needed to be added to the modeling equation for the

best fit curve, much like the larger gap sizes of the gap experiment. This linear variable

can be attributed to the velocity dependent friction from the piston rod and seals within

the cylinder. It is not visible in the larger currents primarily because the larger forces

from the magnetic valve begin to “drown out” the friction forces. Figure 17 shows the

best fit curves for each of the applied currents for both the compression and extension

strokes. The scatter plot data has been eliminated from this plot for readability. This plot

shows that at a speed of 40 mm/s, the resistance of the damper increases by 161% when a

current of 2.083 A is applied to the magnets.

This increase in resistance is quite evident to the user due to the fact that at this

applied current, the resistance severely dampens the cylinder, almost to the point of

“locking up” the cylinder. Also notice how the constant offsets on each the best fit

curves increase as the current increases. This increase in the constant offset is expected

since as the magnetic field strength increases, a stronger static pressure must initially

build up to displace the magnet from the piston head and allow fluid to flow through the

orifice. These offsets correlate with the constant offsets that needed to be added to the

smaller gap data for the gap experiment. Finally, the groupings of the best fit curves at

low currents and high currents fall exactly how one would expect functions of magnetic

fields to plot. When dealing with electromagnets, the strength of the magnetic field will

often rise in an exponential fashion.

32

Figure 15: Force-Velocity Characteristics from Applied Current for Clear Cylinder - Least square best fit curves for the extension and compression strokes of the clear damper for each applied current.

Figure 16: Sample Force-Velocity data for the Magnetic Strength Experiment - Force-velocity data for I = 260mA and the corresponding least square best fit curve.

33

Figure 17: Force-Velocity Characteristics from Applied Current for Steel Cylinder - Least square best fit curves for the extension and compression strokes of the steel damper for each applied current.

Conversely, as the magnets begin to reach the maximum magnetic field they are

able to produce, often referred to as saturation, the magnetic field strength begins to

decrease in an exponential fashion. This activity is seen in this plot.

Comparison of Steel Cylinder and Clear Cylinder Data

The hypothesis function, )(xA , showed to followed the data for the clear cylinder

relatively well; even though 3 of the Q(x) values fell close to the value of Qmin.

Therefore, the piston head was redesigned for the steel cylinder with a smaller orifice in

order to test in a larger force range. The author hoped the larger force range would

34

produce more definitive values of Q(x) for the new cylinder. Also, more testing was

conducted with gap sizes that fell on the quadratic curve, Q(x), of the clear cylinder. The

hypothesis function of the gap testing cylinder did not follow the Q(x) data of the gap

testing piston as well as the hypothesis function of the clear cylinder followed its

respective data. Several suggestions were mentioned regarding the behavior of the gap

testing cylinder. In redesigning the damper for the steel cylinder, it was desirable to

increase the range of the damper resistance due to the applied current. Therefore, the

length of the electromagnets was slightly increased and a larger gauge of wire was used

for the winding. The larger gauge wire lowered the winding resistance to allow for more

current. Increasing current would, in turn, increase the strength of the magnetic field

force. Upon testing, the larger magnets significantly increased the resistance range of the

steel cylinder from that of the clear cylinder.

Even though the hypothesis effective orifice area for the testing piston did not

follow values of Q(x) as well as the hypothesis function for the clear damper, the

implementation and examination of the testing piston proved that the hypothesis has

some validity. Also, the redesign of the magnets helped to significantly increase the

resistance of the steel damper over the clear damper. This increase in resistance makes

the steel cylinder more usable in an exercising situation.

Conclusions

This thesis presents a design for an adjustable damper that can be implemented

into exercise equipment. The design satisfies the design requirements of simplicity and

internal, dynamic resistance control. After examining current adjustable damper and

35

fluid technologies, it was decided that dampers in exercise equipment would benefit the

most from an adjustable orifice design.

Using two oppositely poled electromagnets, a magnetic valve has been designed

and described. Analysis of the magnetic valve showed a significant correlation between

the size of the gap separating the magnet and piston head, and the effective size of the

orifices on the piston head. The hypothesis function of the magnetic valve matched

reasonably well with the presented force-velocity of the two different dampers. The

hypothesis function of the clear damper correlated with its respective data the better than

the hypothesis function of the steel damper.

Finally, the force-velocity characteristics for a range of applied currents were also

analyzed to demonstrate the strength characteristics of the two developed dampers. The

characteristics of the clear damper closely followed the theoretical expectation, where the

steel damper acted as expected. In addition, the testing and analysis showed that the steel

damper provided a wider range of resistance control depending on the applied current.

At a maximum current value, the damper became quite difficult to move for exercisers.

If recent trends continue, fluid damper resistance for exercise equipment will

become more and more prevalent. This thesis presents solutions for internal resistance

adjustability, although future work is still needed. The relationship between the applied

current and the gap size still need to be quantified in order to provide a damper that can

be dynamically controlled using feedback.

The following appendices provide code used to collect and process the data for

each experiment. Also, the scatter plots for each of the data sets in the experiments are

provided with the corresponding least square error best fit curve.

36

APPENDICES

37

APPENDIX A

MATLAB Code – Gap data

Readin.m

%This function reads in data column values from Excel spreadsheets %and loads them into corresponding vectors force_018raw=dlmread('018forcedata.xls','\t'); vel_018raw=dlmread('018veldata.xls','\t'); press_018raw=dlmread('018pressdata.xls','\t'); force_027raw=dlmread('027forcedata.xls','\t'); vel_027raw=dlmread('027veldata.xls','\t'); press_027raw=dlmread('027pressdata.xls','\t'); force_036raw=dlmread('036forcedata.xls','\t'); vel_036raw=dlmread('036veldata.xls','\t'); press_036raw=dlmread('036pressdata.xls','\t'); force_045raw=dlmread('045forcedata.xls','\t'); vel_045raw=dlmread('045veldata.xls','\t'); press_045raw=dlmread('045pressdata.xls','\t'); force_060raw=dlmread('060forcedata.xls','\t'); vel_060raw=dlmread('060veldata.xls','\t'); press_060raw=dlmread('060pressdata.xls','\t'); force_084raw=dlmread('084forcedata.xls','\t'); vel_084raw=dlmread('084veldata.xls','\t'); press_084raw=dlmread('084pressdata.xls','\t'); force_120raw=dlmread('120forcedata.xls','\t'); vel_120raw=dlmread('120veldata.xls','\t'); press_120raw=dlmread('120pressdata.xls','\t');

38

process.m

%This function calls the function 'convert' and places the returned vectors %into corresponding vector names [force_018, vel_018] = convert(force_018raw, vel_018raw); [force_027, vel_027] = convert(force_027raw, vel_027raw); [force_036, vel_036] = convert(force_036raw, vel_036raw); [force_045, vel_045] = convert(force_045raw, vel_045raw); [force_060, vel_060] = convert(force_060raw, vel_060raw); [force_084, vel_084] = convert(force_084raw, vel_084raw); [force_120, vel_120] = convert(force_120raw, vel_120raw);

39

convert.m

%This function takes the passed in force and velocity vectors and crops %them to the same size. It also converts the vectors into metric units %before returning them function [force_out,vel_out] = convert(force_in,vel_in) forcelength = length(force_in); vellength = length(vel_in); if(vellength>forcelength) %compare to see which vector is longer vel_out(:,2) = vel_in(1:forcelength,2); %cropping longer vector force_out = force_in; end if(forcelength>vellength) force_out(:,2) = force_in(1:vellength,2); vel_out = vel_in; end vel_out(:,2) = 0.0254*vel_out(:,2); %converting to meters force_out(:,2) = 4.44822162*100*force_out(:,2); %converting to Newtons

40

get_ext_stroke.m

%This function calls the function 'ext_stroke' for each of the data sets [force_018_ext, vel_018_ext] = ext_stroke(force_018,vel_018); [force_027_ext, vel_027_ext] = ext_stroke(force_027,vel_027); [force_036_ext, vel_036_ext] = ext_stroke(force_036,vel_036); [force_045_ext, vel_045_ext] = ext_stroke(force_045,vel_045); [force_060_ext, vel_060_ext] = ext_stroke(force_060,vel_060); [force_084_ext, vel_084_ext] = ext_stroke(force_084,vel_084); [force_120_ext, vel_120_ext] = ext_stroke(force_120,vel_120);

41

ext_stroke.m

%This function reads in complete force and velocity vectors and finds the %data points corresponding to the extension stroke (positive force and velocity %together). A new vector is created with these data points and returned. function [force_out, vel_out] = ext_stroke(force_in, vel_in) index = 6; %index for syncronizing force and velocity data ind = find(force_in(:,2) > 0); %find indices of positive force for i=1:length(ind), %build vectors for positive indices force_out(i) = force_in(ind(i),2); if((ind(i) + index) > length(vel_in)) vel_out(i) = vel_in(ind(i),2); else vel_out(i) = vel_in(ind(i) + index,2); end end

42

avals.m

%This function calls function 'geta' and uses the returned values to plot %against the gap distances a1 = geta(force_018_ext,vel_018_ext); %function call to obtain a values mean_a1 = mean(medfilt1(a1,500)) %filters the data to remove ambient data a2 = geta(force_027_ext,vel_027_ext); mean_a2 = mean(medfilt1(a2,500)) a3 = geta(force_036_ext,vel_036_ext); mean_a3 = mean(medfilt1(a3,500)) a4 = geta(force_045_ext,vel_045_ext); mean_a4 = mean(medfilt1(a4,500)) a5 = geta(force_060_ext,vel_060_ext); mean_a5 = mean(medfilt1(a5,500)) a6 = geta(force_084_ext,vel_084_ext); mean_a6 = mean(medfilt1(a6,500)) a7 = geta(force_120_ext,vel_120_ext); mean_a7 = mean(medfilt1(a7,500)) mean_a = [mean_a2 mean_a3 mean_a4 mean_a5 mean_a6 mean_a7]; x = [0.27 0.36 0.45 0.60 0.84 1.2]; plot(x, mean_a,'.','markersize',20) %plots the averaged a values against the gap distances %axis([0.2 1.4 0.09 0.18]);

43

geta.m

%This function calculates and returns the a value for each of the data %points within the force and velocity data sets. function a = geta(force_in,vel_in) for i=1:length(vel_in) a(i) = force_in(i)/(vel_in(i)*vel_in(i)); %calculate a from equation f = a*v^2 end

44

Qvals.m

%This function calculates Bernoulli's equation for the given orifice size %and calculates the modeling line. vec_in is a vector containing the %mean_a values for the data sets. This function then plots the Bernoulli %line on the same plot as the a versus gap size values. function Qvals(mean_a,s) x = [0.27 0.36 0.45 0.60 0.84 1.2]; %vector of gap sizes orifice = 2.4892; %orifice diameter in mm k = 0.9; %constant in Bernoulli's eq. p = 843; %density of the fluid Ap = pi*(45.72/2000)^2; %area of the piston head Ain = pi*(orifice/2000)^2; %area of the orifice tempx = linspace(.23,1.25,250); for i=1:length(tempx) temp = 2*pi*(orifice/2000)*(tempx(i)/1000); %calculation of input area if (temp < Ain) %comparison between calculated input area and maximum input area Q(i) = (p*Ap^3)/(2*k^2*temp^2); %calculation of Bernoulli's equation else Q(i) = (p*Ap^3)/(2*k^2*Ain^2); end end Q(240) plot(x,mean_a,'.','markersize',20); %plot of a values versus gap sizes hold on axis([0.2 1.25 0.8*10^5 1.8*10^5]); plot(tempx, Q) %plot of Bernoulli's equation xlabel('gap x (mm)'); ylabel('Q'); x2 = linspace(0,2,250); hold on plot(x2,4250./(x2.*x2)+97230,'--'); line([.27 .27], [(mean_a(1) - s(2)) (mean_a(1) + s(2))]); %plot vertical deviation line line([.36 .36], [(mean_a(2) - s(3)) (mean_a(2) + s(3))]);

45

line([.45 .45], [(mean_a(3) - s(4)) (mean_a(3) + s(4))]); line([.60 .60], [(mean_a(4) - s(5)) (mean_a(4) + s(5))]); line([.84 .84], [(mean_a(5) - s(6)) (mean_a(5) + s(6))]); line([1.20 1.20], [(mean_a(6) - s(7)) (mean_a(6) + s(7))]); hold off legend('Q(x) measured','Effective orifice area(hypothesis)','Effective orifice area(measured)');

46

plotFV.m

%This function plots the force/velocity characteristics for each of the %different gap size data sets. The plots are in subplot format. index = 5; %value used to syncronize data subplot(2,2,1); plot(vel_027(3+index:length(vel_027),2),force_027(3:length(vel_027)-index,2),'.'); %plot raw data hold on plot(x,.15*x.^2+0.14*x+0.5, 'k', 'linewidth', 2) %plot best-fit line hold off title('0.25mm gap'); axis([0 4 0 2]); subplot(2,2,2); plot(vel_036(3+index:length(vel_036),2),force_036(3:length(vel_036)-index,2),'.'); hold on plot(x,.13*x.^2+0.11*x+0.5, 'k', 'linewidth', 2) hold off title('0.37mm gap'); axis([0 4 0 2]); subplot(2,2,3); plot(vel_045(3+index:length(vel_045),2),force_045(3:length(vel_045)-index,2),'.'); hold on plot(x,.11*x.^2+0.10*x+0.5, 'k', 'linewidth', 2) hold off title('0.51mm gap'); axis([0 4 0 2]); index = 5; subplot(2,2,4); plot(vel_060(3+index:length(vel_060),2),force_060(3:length(vel_060)-index,2),'.'); hold on plot(x,.095*x.^2+0.10*x+0.5, 'k', 'linewidth', 2) hold off title('0.66mm gap');

47

axis([0 4 0 2]); figure(2) subplot(2,2,1); plot(vel_084(3+index:length(vel_084),2),force_084(3:length(vel_084)-index,2),'.'); hold on plot(x,.0875*x.^2+0.10*x+0.5, 'k', 'linewidth', 2) hold off title('0.84mm gap'); axis([0 4 0 2]); subplot(2,2,2); plot(vel_120(3+index:length(vel_120),2),force_120(3:length(vel_120)-index,2),'.'); hold on plot(x,.11*x.^2+0.10*x+0.5, 'k', 'linewidth', 2) hold off title('1.02mm gap'); axis([0 4 0 2]); subplot(2,2,4); plot(x,.15*x.^2+0.14*x+0.5, 'r', 'linewidth', 2) hold on plot(x,.13*x.^2+0.11*x+0.5, 'g', 'linewidth', 2) plot(x,.11*x.^2+0.10*x+0.5, 'b', 'linewidth', 2) plot(x,.095*x.^2+0.10*x+0.5, 'y', 'linewidth',2) plot(x,.0875*x.^2+0.10*x+0.5, 'm', 'linewidth',2) plot(x,.12*x.^2+0.08*x+0.5, 'k', 'linewidth', 2) hold off

48

plotall.m

%This function plots all of the force-velocity data for the gap experiment %in scatter plot form with different colors representing different data %sets index = 6; hold on; axis([0 6 0 4]); plot(vel_027(3+index:length(vel_027),2),force_027(3:length(vel_027)-index,2),'.','markeredgecolor','k'); pause(1); plot(vel_036(3+index:length(vel_036),2),force_036(3:length(vel_036)-index,2),'.','markeredgecolor','r'); pause(1); plot(vel_045(3+index:length(vel_045),2),force_045(3:length(vel_045)-index,2),'.','markeredgecolor','g'); pause(1); plot(vel_060(3+index:length(vel_060),2),force_060(3:length(vel_060)-index,2),'.','markeredgecolor','y'); pause(1); plot(vel_084(3+index:length(vel_084),2),force_084(3:length(vel_084)-index,2),'.','markeredgecolor','m'); pause(1); plot(vel_120(3+index:length(vel_120),2),force_120(3:length(vel_120)-index,2),'.','markeredgecolor','b');

49

stdev.m

%This function takes the given a values for the gap experiment and finds %the corresponding standard deviation. A vertical line is then plotted on %top of the points to represent the range of the standard deviation tempa1 = medfilt1(a1,500); %filter incoming data s1 = std(tempa1); %compute standard deviation tempa2 = medfilt1(a2,500); s2 = std(tempa2); tempa3 = medfilt1(a3,500); s3 = std(tempa3); tempa4 = medfilt1(a4,500); s4 = std(tempa4); tempa5 = medfilt1(a5,500); s5 = std(tempa5); tempa6 = medfilt1(a6,500); s6 = std(tempa6); tempa7 = medfilt1(a7,500); s7 = std(tempa7); sd = [s1 s2 s3 s4 s5 s6 s7] x2 = linspace(0,2,250); hold on plot(x2,4250./(x2.*x2)+97230,'--'); line([.27 .27], [(mean_a2 - s2) (mean_a2 + s2)]); %plot vertical deviation line line([.36 .36], [(mean_a3 - s3) (mean_a3 + s3)]); line([.45 .45], [(mean_a4 - s4) (mean_a4 + s4)]); line([.60 .60], [(mean_a5 - s5) (mean_a5 + s5)]); line([.84 .84], [(mean_a6 - s6) (mean_a6 + s6)]); line([1.20 1.20], [(mean_a7 - s7) (mean_a7 + s7)]); hold off

50

APPENDIX B

MATLAB Code – Force-velocity Data from Applied Current

readin.m

%This function reads in data column values from Excel spreadsheets for the %applied current experiment and loads them into corresponding vectors force_0000raw=dlmread('0000forcedata.xls','\t'); vel_0000raw=dlmread('0000veldata.xls','\t'); press_0000raw=dlmread('0000pressdata.xls','\t'); force_0260raw=dlmread('0260forcedata.xls','\t'); vel_0260raw=dlmread('0260veldata.xls','\t'); press_0260raw=dlmread('0260pressdata.xls','\t'); force_0521raw=dlmread('0521forcedata.xls','\t'); vel_0521raw=dlmread('0521veldata.xls','\t'); press_0521raw=dlmread('0521pressdata.xls','\t'); force_0781raw=dlmread('0781forcedata.xls','\t'); vel_0781raw=dlmread('0781veldata.xls','\t'); press_0781raw=dlmread('0781pressdata.xls','\t'); force_1042raw=dlmread('1042forcedata.xls','\t'); vel_1042raw=dlmread('1042veldata.xls','\t'); press_1042raw=dlmread('1042pressdata.xls','\t'); force_1302raw=dlmread('1302forcedata.xls','\t'); vel_1302raw=dlmread('1302veldata.xls','\t'); press_1302raw=dlmread('1302pressdata.xls','\t'); force_1563raw=dlmread('1563forcedata.xls','\t'); vel_1563raw=dlmread('1563veldata.xls','\t'); press_1563raw=dlmread('1563pressdata.xls','\t'); force_1823raw=dlmread('1823forcedata.xls','\t'); vel_1823raw=dlmread('1823veldata.xls','\t'); press_1823raw=dlmread('1823pressdata.xls','\t'); force_2083raw=dlmread('2083forcedata.xls','\t'); vel_2083raw=dlmread('2083veldata.xls','\t'); press_2083raw=dlmread('2083pressdata.xls','\t');

51

process.m

%This function calls the function 'convert' and places the returned vectors %into corresponding vector names [force_0000, vel_0000] = convert(force_0000raw, vel_0000raw); [force_0260, vel_0260] = convert(force_0260raw, vel_0260raw); [force_0521, vel_0521] = convert(force_0521raw, vel_0521raw); [force_0781, vel_0781] = convert(force_0781raw, vel_0781raw); [force_1042, vel_1042] = convert(force_1042raw, vel_1042raw); [force_1302, vel_1302] = convert(force_1302raw, vel_1302raw); [force_1563, vel_1563] = convert(force_1563raw, vel_1563raw); [force_1823, vel_1823] = convert(force_1823raw, vel_1823raw); [force_2083, vel_2083] = convert(force_2083raw, vel_2083raw);

52

convert.m

%This function takes the passed in force and velocity vectors and crops %them to the same size. It also converts the vectors into metric units %before returning them function [force_out,vel_out] = convert(force_in,vel_in) forcelength = length(force_in); vellength = length(vel_in); if(vellength>forcelength) %compare to see which vector is longer vel_out(:,2) = vel_in(1:forcelength,2); %cropping longer vector force_out = force_in; end if(forcelength>vellength) force_out(:,2) = force_in(1:vellength,2); vel_out = vel_in; end %vel_out(:,2) = 0.0254*vel_out(:,2); %converting to meters vel_out(:,2) = 25.4*vel_out(:,2); %converting to millimeters force_out(:,2) = 4.44822162*100*force_out(:,2); %converting to Newtons

53

get_ext_stroke.m

%This function calls the function 'ext_stroke' for each of the data sets [force_0000_ext, vel_0000_ext] = ext_stroke(force_0000,vel_0000); [force_0260_ext, vel_0260_ext] = ext_stroke(force_0260,vel_0260); [force_0521_ext, vel_0521_ext] = ext_stroke(force_0521,vel_0521); [force_0781_ext, vel_0781_ext] = ext_stroke(force_0781,vel_0781); [force_1042_ext, vel_1042_ext] = ext_stroke(force_1042,vel_1042); [force_1302_ext, vel_1302_ext] = ext_stroke(force_1302,vel_1302); [force_1563_ext, vel_1563_ext] = ext_stroke(force_1563,vel_1563); [force_1823_ext, vel_1823_ext] = ext_stroke(force_1823,vel_1823); [force_2083_ext, vel_2083_ext] = ext_stroke(force_2083,vel_2083);

54

ext_stroke.m

%This function reads in complete force and velocity vectors from the %applied current experiments and finds the data points corresponding to the %extension stroke (positive force and velocity together). A new vector is %created with these data points and returned. function [force_out, vel_out] = ext_stroke(force_in, vel_in) index = 6; %index for syncronizing force and velocity data ind = find(force_in(:,2) > 0); %find indices of positive force for i=1:length(ind), %build vectors for positive indices force_out(i) = force_in(ind(i),2); if((ind(i) + index) > length(vel_in)) vel_out(i) = vel_in(ind(i),2); else vel_out(i) = vel_in(ind(i) + index,2); end end

55

avals.m

%This function calls function 'geta' and uses the returned values to plot %the best fit curve to represent the FV characteristics for a given applied %current a1 = geta(force_0000_ext,vel_0000_ext); %function call to obtain a values mean_a1 = mean(medfilt1(a1,500)) %filters the data to remove ambient data a2 = geta(force_0260_ext,vel_0260_ext); mean_a2 = mean(medfilt1(a2,500)) a3 = geta(force_0521_ext,vel_0521_ext); mean_a3 = mean(medfilt1(a3,500)) a4 = geta(force_0781_ext,vel_0781_ext); mean_a4 = mean(medfilt1(a4,500)) a5 = geta(force_1042_ext,vel_1042_ext); mean_a5 = mean(medfilt1(a5,500)) a6 = geta(force_1302_ext,vel_1302_ext); mean_a6 = mean(medfilt1(a6,500)) a7 = geta(force_1563_ext,vel_1563_ext); mean_a7 = mean(medfilt1(a7,500)) a8 = geta(force_1823_ext,vel_1823_ext); mean_a8 = mean(medfilt1(a8,500)) a9 = geta(force_2083_ext,vel_2083_ext); mean_a9 = mean(medfilt1(a9,500)) x = linspace(0,0.06,400); plot(x,mean_a1.*x.*x); hold on axis([-0.04 0.04 -2000 2000]); plot(x,mean_a2.*x.*x); %plots the best fit curve for the FV data at the given applied current plot(x,mean_a3.*x.*x); plot(x,mean_a4.*x.*x); plot(x,mean_a5.*x.*x); plot(x,mean_a6.*x.*x); plot(x,mean_a7.*x.*x); plot(x,mean_a8.*x.*x); plot(x,mean_a9.*x.*x); plot(-x,-mean_a1.*x.*x);

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plot(-x,-mean_a2.*x.*x); plot(-x,-mean_a3.*x.*x); plot(-x,-mean_a4.*x.*x); plot(-x,-mean_a5.*x.*x); plot(-x,-mean_a6.*x.*x); plot(-x,-mean_a7.*x.*x); plot(-x,-mean_a8.*x.*x); plot(-x,-mean_a9.*x.*x); % mean_a = [mean_a1 mean_a2 mean_a3 mean_a4 mean_a5 mean_a6 mean_a7 mean_a8 mean_a9]; % x = [0.27 0.36 0.45 0.60 0.84 1.2]; % % plot(x, mean_a,'.','markersize',20) %axis([0.2 1.4 0.09 0.18]);

57

geta.m

%This function calculates and returns the a value for each of the data %points within the force and velocity data sets from the applied current %experiment. function a = geta(force_in,vel_in) for i=1:length(vel_in) a(i) = force_in(i)/(vel_in(i)*vel_in(i)); %calculate a from equation f = a*v^2 end

58

FI_plots.m

%This function plots the best fit curve calculated for each of the %different applied current data sets. The units of the plot are m vs. N x = linspace(0,75,500); plot(x,3.2281*10^5.*x.*x+3000*x+120); hold on; axis([-0.06 0.06 -2000 2000]); plot(x,3.5281*10^5.*x.*x+3000*x+120); plot(x,3.7281*10^5.*x.*x+1500*x+150); plot(x,3.8281*10^5.*x.*x+1500*x+170); plot(x,3.4281*10^5.*x.*x+3500*x+190); plot(x,3.9281*10^5.*x.*x+3500*x+240); plot(x,4.1281*10^5.*x.*x+3500*x+320); plot(x,4.9281*10^5.*x.*x+3500*x+340); plot(x,5.2281*10^5.*x.*x+4000*x+350); plot(-x,-3.2281*10^5.*x.*x+3000*x+120); plot(-x,-3.5281*10^5.*x.*x+3000*x+120); plot(-x,-3.7281*10^5.*x.*x+1500*x+150); plot(-x,-3.8281*10^5.*x.*x+1500*x+170); plot(-x,-3.4281*10^5.*x.*x+3500*x+190); plot(-x,-3.9281*10^5.*x.*x+3500*x+240); plot(-x,-4.1281*10^5.*x.*x+3500*x+320); plot(-x,-4.9281*10^5.*x.*x+3500*x+340); plot(-x,-5.2281*10^5.*x.*x+4000*x+350); temp = [3.5281*10^5 3.7281*10^5 3.8281*10^5 3.4281*10^5 3.9281*10^5 4.1281*10^5 4.9281*10^5 5.2281*10^5]; i = [260 521 781 1042 1302 1563 1823 2083]; plot(i,temp,'.')

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FI_plots_mm.m

%This function plots the best fit curve calculated for each of the %different applied current data sets. The units of the plot are mm vs. N x = linspace(0,75,500); plot(x,0.275.*x.*x+3*x+250, 'b', 'linewidth', 2) hold on axis([-70 70 -2000 2000]); plot(x,0.31.*x.*x+3*x+250, 'b', 'linewidth', 2) plot(x,0.32.*x.*x+3*x+250, 'b', 'linewidth', 2) plot(x,0.33.*x.*x+360, 'b', 'linewidth', 2) plot(x,0.345.*x.*x+360, 'b', 'linewidth', 2) plot(x,0.375.*x.*x+450, 'b', 'linewidth', 2) plot(x,0.405.*x.*x+520, 'b', 'linewidth', 2) plot(x,0.49.*x.*x+520, 'b', 'linewidth', 2) plot(x,0.52.*x.*x+520, 'b', 'linewidth', 2) plot(-x,-0.275.*x.*x+3*x+250, 'b', 'linewidth', 2) plot(-x,-0.31.*x.*x+3*x+250, 'b', 'linewidth', 2) plot(-x,-0.32.*x.*x+3*x+250, 'b', 'linewidth', 2) plot(-x,-0.33.*x.*x+360, 'b', 'linewidth', 2) plot(-x,-0.345.*x.*x+360, 'b', 'linewidth', 2) plot(-x,-0.375.*x.*x+450, 'b', 'linewidth', 2) plot(-x,-0.405.*x.*x+520, 'b', 'linewidth', 2) plot(-x,-0.49.*x.*x+520, 'b', 'linewidth', 2) plot(-x,-0.52.*x.*x+520, 'b', 'linewidth', 2) xlabel('Velocity (mm/s)'); ylabel('Force (Newtons)');

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plotall.m

%This function plots all of the force-velocity data for the applied current %experiment in scatter plot form with different colors representing %different data sets index = 6; plot(vel_0000_ext(3+index:length(vel_0000_ext)),force_0000_ext(3:length(vel_0000_ext)-index),'.','markeredgecolor','b'); hold on; axis([0 70 0 2000]); pause(1); plot(vel_0260_ext(3+index:length(vel_0260_ext)),force_0260_ext(3:length(vel_0260_ext)-index),'.','markeredgecolor','k'); pause(1); plot(vel_0521_ext(3+index:length(vel_0521_ext)),force_0521_ext(3:length(vel_0521_ext)-index),'.','markeredgecolor','r'); pause(1); plot(vel_0781_ext(3+index:length(vel_0781_ext)),force_0781_ext(3:length(vel_0781_ext)-index),'.','markeredgecolor','g'); pause(1); plot(vel_1042_ext(3+index:length(vel_1042_ext)),force_1042_ext(3:length(vel_1042_ext)-index),'.','markeredgecolor','y'); pause(1); plot(vel_1302_ext(3+index:length(vel_1302_ext)),force_1302_ext(3:length(vel_1302_ext)-index),'.','markeredgecolor','m'); pause(1); plot(vel_1563_ext(3+index:length(vel_1563_ext)),force_1563_ext(3:length(vel_1563_ext)-index),'.','markeredgecolor','b'); pause(1); plot(vel_1823_ext(3+index:length(vel_1823_ext)),force_1823_ext(3:length(vel_1823_ext)-index),'.','markeredgecolor','k'); pause(1); plot(vel_2083_ext(3+index:length(vel_2083_ext)),force_2083_ext(3:length(vel_2083_ext)-index),'.','markeredgecolor','r');

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separate_extension_plots.m

%This function plots the force-velocity characteristics for each of the %different applied current data sets. The plots are each in a different %figure. figure(1) index = 6; plot(vel_0000_ext(3+index:length(vel_0000_ext)),force_0000_ext(3:length(vel_0000_ext)-index),'.','markeredgecolor','k'); hold on; axis([0 70 0 2000]); plot(x,0.275.*x.*x+3*x+250, 'b', 'linewidth', 2) xlabel('Velocity (mm/s)'); ylabel('Force (Newtons)'); hold off; figure(2) plot(vel_0260_ext(3+index:length(vel_0260_ext)),force_0260_ext(3:length(vel_0260_ext)-index),'.','markeredgecolor','k'); hold on; axis([0 70 0 2000]); plot(x,0.31.*x.*x+3*x+250, 'b', 'linewidth', 2) xlabel('Velocity (mm/s)'); ylabel('Force (Newtons)'); hold off figure(3) plot(vel_0521_ext(3+index:length(vel_0521_ext)),force_0521_ext(3:length(vel_0521_ext)-index),'.','markeredgecolor','k'); hold on; axis([0 70 0 2000]); plot(x,0.32.*x.*x+3*x+250, 'b', 'linewidth', 2) xlabel('Velocity (mm/s)'); ylabel('Force (Newtons)'); hold off figure(4) plot(vel_0781_ext(3+index:length(vel_0781_ext)),force_0781_ext(3:length(vel_0781_ext)-index),'.','markeredgecolor','k'); hold on;

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axis([0 70 0 2000]); plot(x,0.33.*x.*x+360, 'b', 'linewidth', 2) xlabel('Velocity (mm/s)'); ylabel('Force (Newtons)'); hold off figure(5) plot(vel_1042_ext(3+index:length(vel_1042_ext)),force_1042_ext(3:length(vel_1042_ext)-index),'.','markeredgecolor','k'); hold on; axis([0 70 0 2000]); plot(x,0.345.*x.*x+360, 'b', 'linewidth', 2) xlabel('Velocity (mm/s)'); ylabel('Force (Newtons)'); hold off figure(6) plot(vel_1302_ext(3+index:length(vel_1302_ext)),force_1302_ext(3:length(vel_1302_ext)-index),'.','markeredgecolor','k'); hold on; axis([0 70 0 2000]); plot(x,0.375.*x.*x+450, 'b', 'linewidth', 2) xlabel('Velocity (mm/s)'); ylabel('Force (Newtons)'); hold off figure(7) plot(vel_1563_ext(3+index:length(vel_1563_ext)),force_1563_ext(3:length(vel_1563_ext)-index),'.','markeredgecolor','k'); hold on; axis([0 70 0 2000]); plot(x,0.405.*x.*x+520, 'b', 'linewidth', 2) xlabel('Velocity (mm/s)'); ylabel('Force (Newtons)'); hold off figure(8) plot(vel_1823_ext(3+index:length(vel_1823_ext)),force_1823_ext(3:length(vel_1823_ext)-index),'.','markeredgecolor','k'); hold on; axis([0 70 0 2000]); plot(x,0.49.*x.*x+520, 'b', 'linewidth', 2) xlabel('Velocity (mm/s)'); ylabel('Force (Newtons)'); hold off

63

figure(9) plot(vel_2083_ext(3+index:length(vel_2083_ext)),force_2083_ext(3:length(vel_2083_ext)-index),'.','markeredgecolor','k'); hold on; axis([0 70 0 2000]); plot(x,0.52.*x.*x+520, 'b', 'linewidth', 2) xlabel('Velocity (mm/s)'); ylabel('Force (Newtons)'); hold off

64

APPENDIX C

LabVIEW Data Acquisition Code

MagThesis.vi Front Panel

65

MagThesis.vi Block Diagram

66

67

68

69

70

71

hkWriteLong.vi Front Panel

hkWriteLong.vi Block Diagraml

72

APPENDIX D

HyperKernel C Code

CurvesDevelop.c // Curvesdevelop.C Ian Gravagne, Brett Levins 3/17/04, Baylor University // // A linear displacement and velocity test routine. #include <hKernel.h> #include "C:\Documents and Settings\levins\Desktop\Realtime\\stgdriver\\stgdriver.h" #include "C:\Documents and Settings\levins\Desktop\Realtime\\databuffer\\databuffer.h" void analog_in_thread(void); void analog_in_thread_ISR(void); long analog_in_thread_ID; long *buffer; long *buffer2; long *buffer3; double low_bound(double); double g(double a, double b); double signum(double); void main(void) { double zero[] = {0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0}; double ain[8]; // Print some instructions hkPrint(""); hkPrint("---------------------"); hkPrint("Linear Displacement and Velocity Reading System (Ian Gravagne, Brett Levins, 2004)"); hkPrint("---------------------"); hkPrint(""); // Do some initialization stgSetBaseAddress(0x300);

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stgAnalogCalibrate(); stgAnalogAllOut(zero); stgInitEncoders(); stgSetPortDirection(STG_PORT_A, OUTPUT); buffer = bufCreate("Position", 2000); buffer2 = bufCreate("Velocity", 2000); buffer3 = bufCreate("Force", 2000); // Start ADC waveform thread and attach the timer ISR analog_in_thread_ID = hkCreateThread(analog_in_thread, MIN_STACK_SIZE, "analog_in", 28); hkAttachSystemTimer(analog_in_thread_ISR, NULL); // Loop until stopped... while(!hkQuitRequest()) { // Read state of ADC channels stgAnalogAllIn(ain); // delay for a while hkSleep(100); } // cleanup, and zero out the DACs hkDetachSystemTimer(analog_in_thread_ISR); hkKillThread(analog_in_thread_ID); stgAnalogAllOut(zero); stgDigitalAllOut(0x00); bufDestroy("Position"); bufDestroy("Velocity"); bufDestroy("Force"); } void analog_in_thread(void) { long i = 0; // sample counter long j = 0; // timer counter long k = 0; int count = 0; int state = 0; int angle = 0; int flag; double x = 0.0;

74

double fs = 1000.0; // sampling frequency double force; double pressure, pressurea, pressureb; double position, position2 = 0.0, position3 = 0.0, position4 = 0.0, position5 = 0.0; double velocity, velocity2 = 0.0, velocity3 = 0.0, velocity4 = 0.0, velocity5 = 0.0; double accel, accel2 = 0.0, accel3 = 0.0, accel4 = 0.0, accel5 = 0.0; double heartrate; double currenti = 0.0, currenti2 = 0.0, currenti3 = 0.0, currenti4 = 0.0; double currento = 0.0, currento2 = 0.0, currento3 = 0.0, currento4 = 0.0; double voltage; double magnet1, magnet2, magnets; double MA; double dir=0; while(1) { if(hkReceive(ANYTID,NULL,0) == SIGNALED) { if(j%20 == 0) { i++; magnet1 = (double)hkReadLong(1)/1000.0; magnet2 = (double)hkReadLong(0)/1000.0; magnets = (double)hkReadLong(2)/1000.0; if (magnets > 0.01){ stgAnalogOut(0,magnets); stgAnalogOut(1,magnets); } else if (magnets < 0.01){ stgAnalogOut(0,magnet1); stgAnalogOut(1,magnet2); } else{ stgAnalogOut(0,0.0); stgAnalogOut(1,0.0); } position = -(long)stgReadEncoder(0)/205.0;

75

force = stgAnalogIn(1); velocity = 1.4816*velocity2 - 0.5488*velocity3 + 6.6674*position2 - 6.6674*position3; accel = 1.4816*accel2 - 0.5488*accel3 + 6.6674*velocity2 - 6.6674*velocity3; position5 = position4;position4 = position3;position3 = position2;position2 = position; velocity5 = velocity4;velocity4 = velocity3;velocity3 = velocity2;velocity2 = velocity; accel5 = accel4; accel4 = accel3; accel3 = accel2; accel2 = accel; bufAddPoint(buffer, i*(long)(1000.0*.01), (long)(1000.0*position)); bufAddPoint(buffer2, i*(long)(1000.0*.01), (long)(1000.0*(velocity))); bufAddPoint(buffer3, i*(long)(1000.0*.01), (long)(1000.0*(force + 0.425))); } j++; k++; } } } void analog_in_thread_ISR(void) { hkSignal(analog_in_thread_ID); } double low_bound(double in) { if(in > 0.0) return(in); else

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return(0.0); } double g(double a, double b) { if(fabs(a) > fabs(b)) return(fabs(a-b)); else return(0.0); } double signum(double in) { if(in >= 0.0) return(1.0); else return(-1.0); }

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APPENDIX E

Force-Velocity Data for Gap Experiment

Gap = 0.27mm

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Gap = 0.36mm

Gap = 0.45mm

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Gap = 0.60mm

Gap = 0.84mm

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Gap = 1.20mm

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APPENDIX F

Force-Velocity Data for Applied Current Experiment

Applied Current = 0.0A

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Applied Current = 0.260A

Applied Current = 0.521A

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Applied Current = 0.781A

Applied Current = 1.302A

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Applied Current = 1.563A

Applied Current = 1.823A

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86

BIBLIOGRAPHY

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[2] M.J. Chrzan, J.D. Carlson, “MR fluid sponge devices and their use in vibration

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[8] D.J. Purdy, “Theoretical and experimental investigation into an adjustable

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[10] LORD Materials Division, “What is the Difference between MR and ER Fluid?,”

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[12] B. Levins, I. Gravagne, “A Magnetically Controllable Valve to Vary the Resistance of Hydraulic Dampers for Exercise Equipment,” to appear, Proc. Int. Conf. on Advanced Intelligent Mechatronics (AIM), Monterey, CA, July 2005.